Related papers: Divisor-bounded multiplicative functions in short …

Let $\Lambda(1,n)$ be the $(1,n)$-th Fourier coefficients of $SL(3,\mathbb{Z})$ Hecke-Maass cusp form and $d_3(n)$ denotes the triple divisor function. This paper establishes non-trivial bounds for the averages of these arithmetic functions…

In this paper, we investigate the average behavior of ternary correlations for general $k$-divisor-bounded multiplicative functions, assuming certain second moment integral bounds for the associated $L$-functions. Our approach differs from…

A convenient framework for dealing with asymptotic limit problems of probabilistic nature is provided. These problems include questions such as finding the asymptotic proportion of terms of a sequence falling inside a given interval, or the…

Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued…

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions…

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables…

We provide upper bounds for the mean square integral $$ \int_X^{2X}(\Delta_k(x+h) - \Delta_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $\Delta_k(x)$…

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of…

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture…

Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones…

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from…

In this paper, we develop a general machinery for finding explicit uniform probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the…

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP) for domains in $\mathbb{R}^n$. Recently, the author along with Biswas investigated the problem in the general setting of…

This short note shows a limiting behavior of integrals of some centered antipersistent stationary infinitely divisible moving averages as the compact integration domain in $d\ge 1$ dimensions extends to the whole positive quadrant…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n \le X}f(n)/n$ with error term $O((\log…

We consider the distribution of the binomial probability mass function (pmf) among arithmetic progressions and obtain an average-type theorem. As applications, we consider the possible visits to a kind of sieved sets of integers or lattice…

Let $h\geq 2$. For $A\subseteq \mathbb{N}$ write \[ r_{A,h}(n) := \#\{(x_1,\ldots,x_h)\in A^h ~|~ x_1+\cdots+x_h=n\}. \] We prove a general probabilistic subbasis principle: assuming an asymptotic for a weighted $h$-fold representation sum…

We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting…